truth
Notes on FOL, vi Truth Phil 160 17 November 2016 We’ve already defined for FOL the concepts of possible world and of formula. But the whole point of having either of these things is to say how they fit together: and that is the notion of truth. What makes a formula true in a possible world? Since FOL contains a fair number of moving parts, it will be easiest to proceed in stages. But in brief, the idea is this. A formula says that things are in a certain way. A world determines how things are. At a given world, a formula is true if things are as it says.
1
Atomic formulas
As I’ve urged, formulas which contain free variables represent statements which are incomplete. Therefore, we will define truth only for formulas which are closed. The simplest kind of closed formula consists of an n-place predicate followed by a list of n constants. For example, consider R. Intuitively, this says that a bears to b the relation R. What is it for a world to make that happen? What is it for a world to make things stand in some definite way? Well, recall that a world supplies • a domain, determining which objects exist, and • an extension for each predicate, determining how those things are. For example, a world may look like ∗ : a : W= F : R:
this: {0, 1, 2} 2 {0, 1} {h0, 1i, h1, 0i, h2, 2i}
We want to say that the formula R is true in W, because h0, 1i belongs to the W-extension of R. Likewise F a is false in W because in W, a represents 2 and 2 does not belong to the W-extension of F . 1
Before giving the general definition, a couple of niceties. First, if P is a predicate, I will use this notation: • PW is the W-extension of P. Second, if a is a term, then • aW is the object a stands for in W. Recall that I have been using ordinary numerals 0, 1, 2, . . . to represent numbers, and old-fashioned letters , , to represent their names. We adopt the convention that in every possible world, W = 0 W = 1 W = 2 and so on. Now the definition of truth for closed atomic formulas is this: • A formula Pa is true in W just in case aW belongs to PW . • A formula Qab is true in W just in case (aW , bW ) belongs to QW .
2
Truth-functional combinations
So as you can see, the handling of atomic formulas is pretty straightforward. So is the handling of truth-functions. It carries over exactly from the truth-tables of truth-functional logic. So in this case, the truth or falsehood of a formula is perfectly determined by the truth or falsehood of its simpler parts. Let’s restate the procedure in the new language. Suppose that P and Q are formulas of FOL. Then • W |= ¬P iff W 6|= P • W |= P ∧ Q iff both W |= P and W |= Q; • W |= P ∨ Q iff either W |= P, or W |= Q (or both); • W |= P → Q iff either W 6|= P or W |= Q (or both); • W |= P ↔ Q iff either W |= P and W |= Q, or W 6|= P and W 6|= Q. For example, let W be defined as above. Then have we W |= (F a → F b) ↔ ¬Rba? To settle this question, first apply the definition of truth for closed atomic formulas to determine the truth or falsehood at W of F a, F b, and Rba. Then use the truth-tables to chase the truth-values up the formation tree.
3
Quantification
The treatment of quantification is quite a bit more interesting.1 Consider, for example, the formula ∀xF x. 1 The first solution to this problem was given by Alfred Tarski in 1933, “The concept of truth in formalized languages”. We’ll follow an approach from Joseph Shoenfield’s Mathematical Logic (1967).
2
Our earlier strategy was to reduce the question of its truth-value to the question of truth-values of its simpler parts. However, the only simpler subformula of ∀xF x is F x. And in F x, the x occurs free. This means that F x is analogous to a sentence like “it is green” where the “it” is unspecified. Being not closed, the subformula F x ought not to have a truth-value at all. Thus, quantified formulas are not in general built from simpler closed formulas. So, we need a new idea. Suppose again that the domain of W is {0,1,2}. Then what should W be like in order that it make true a universally quantified formula ∀xA? Intuitively, that formula says “everything has the property expressed by A”, or that every object satisfies A. Now according to W, all the things that exist are simply 0, 1, and 2. So, the formula ∀xA should be true over W provided that each of 0, 1 and 2 satisfies A. So, the question we need to answer is this: • what is it for an object to satisfy a formula over a structure? Well, suppose that A is a formula with the one free variable x. Then, • the object o satisfies A over W provided that the result of plugging in for x a name of o is true in W. For example, 0 satisfies F x just in case F is true, and 2 satisfies ∃y∀z(Rxy → Rxz) just in case ∃y∀z(Ry → Rz) is true. Finally the definition of truth for quantified formulas is this: • ∀xA is true in W iff every element of the domain of W satisfies A. • ∃xA is true in W iff some element of the domain of W satisfies A.
4
The method of truth-functional expansion
In truth-functional logic, the definition of truth is easy to apply: just chase the truth-value up the formation tree. Unfortunately, the situation is not so straightforward in first-order logic. In particular, if the domain is infinite, then the truth-value of a generalization depends on the truth-value of infinitely many simpler formulas.2 Luckily, our main purpose is the assessment of deductive arguments. An argument, as ever, is invalid if it has a countermodel—that is, if there’s a world where all premises are true while the conclusion is false. And it turns out that most invalid arguments you’ll ordinarily encounter have countermodels which are very small, with just two or three elements in their domain. Let’s now develop a practical procedure for evaluating formulas in small structures. The key idea is that once the domain of a structure has been fixed as some finite set of objects, universal and existential quantifiers can be regarded as 2 For example, the natural numbers, together with addition, multiplication, and exponentiation can be considered to form a structure. Problems of classical number theory are simply questions whether a given formula is true in that structure.
3
conjunctions and disjunctions. That is, suppose that the elements of the domain are o1 , . . . , ok . We’ll associate, to each quantified formula A, another formula Ao1 ,...,ok called the truth-functional expansion of A over o1 , . . . , ok . Over any structure whose domain contains just o1 , . . . , ok , the formulas A and Ao1 ,...,ok will have the same truth-value. So over a given finite structure, the method of truth-functional expansion reduces the evaluation of a quantified formula to the evaluation of an unquantified one. To state the method, it will help to have one further piece of terminology. Say that an W-instance of a generalization ∀xA or ∃xA is a formula A[x/o], where o is an element of ∗W . For example, if ∗W contains 1, then an instance of ∃x∃y(F y ∧ F x) is ∃y(F y ∧ F 1). Now the method for computing an expansion of a formula over a domain {o1 , . . . , ok } is simply this. • The expansion of a universal generalization is the conjunction of the expansions of its instances • The expansion of an existential generalization is the disjunction of the expansions of its instances • The expansion of a truth-functional combo of some formulas is just that combo of the expansions of those formulas. • The expansion of an atomic formula is just that formula. Examples. For example, suppose that 0, 1, 2 are the elements of ∗W . And consider a random formula A, like ∃xF x → ∀xF x. The expansion A0,1,2 of A over 0, 1, 2 now gets computed like this: A0,1,2
(∃xF x → ∀xF x)0,1,2 (∃xF x)0,1,2 → (∀xF x)0,1,2 (F 0 ∨ F 1 ∨ F 2) → (F 0 ∧ F 1 ∧ F 2).
We can use the expansion of A to compute the truth-value of A. Suppose, for example, that the W-extension of F is {0, 2}. Granted that ∗W is {0, 1, 2}, this reduces the relevant portion of W to the row of a truth-table! Namely, F0 T
F1 F
F2 . T
So it is clear that W |= F 0 ∨ F 1 ∨ F 2. Meanwhile, W 6|= F 0 ∧ F 1 ∧ F 2. Consequently, W 6|= (F 0 ∨ F 1 ∨ F 2) → (F 0 ∧ F 1 ∧ F 2). 4
And that’s just to say W 6|= (∃xF x → ∀xF x)0,1,2 . Since, however, 0, 1, 2 are precisely the elements of W, it follows that A0,1,2 must, over W, have the same truth-value as A! Therefore, W 6|= ∃xF x → ∀xF x. Caution! It should be clear that first-order logic is much more expressive than truth-functional logic. This is because the quantifiers are a fundamentally new concept. Quantifiers let us say things we can’t say only with truth-functions. Consider, for example, the formula ∀xF x. This is certainly not equivalent to a conjunction of its instances. For no matter how many objects you consider, it’s logically possible for there to exist some other object besides those, and this object might not satisfy F x. Thus, ∀xF x does not follow from any conjunction of atomic formulas. For this reason, it should be clear that a quantified formula is not equivalent to its truth-functional expansion. Rather, the method says only that a formula and its expansion over a domain ∗W have the same truth-value over W. To bring this out, here are a couple of exercises. (i) Find a formula, together with two domains, such that the expansion-of-the-formula-over-one domain is not equivalent to its expansion-over-the-other-domain. (ii) Find a formula and a domain such that the formula does not imply its expansion-over-that-domain. Illustrating the method of expansions. The method of truth-functional expansion is useful for distinguishing the meanings of more complicated formulas. Consider, for example, ∀x∃yRxy and ∃y∀xRxy. Are these equivalent? If so, then they must have the same truth-value over all structures whatsoever. Do they? Consider first a structure whose domain is just {0}. In that case, the two formulas both expand into R00. So they must have the same truth-value over all one-element structures. But consider instead the domain {0, 1}. Then the formulas expand like this. On the one hand, (∀x∃y(Rxy))0,1
(∃y(R0y))0,1 ∧ (∃y(R1y))0,1 (R00 ∨ R01) ∧ (R10 ∨ R11).
On the other hand, (∃y∀x(Rxy))0,1
(∀x(Rx0))0,1 ∨ (∀x(Rx1))0,1 (R00 ∧ R10) ∨ (R01 ∧ R11).
Can you find a structure with domain {0, 1} which makes one true and the other false? Finding such a structure amounts to filling in the question marks in R00 ?
R01 ?
R10 ? 5
R11 ?
This is an exercise in truth-functional logic. You should be able to find a W with ∗W = {0, 1} so that W |= ∀x∃yRxy while W 6|= ∃y∀xRxy.3 This will demonstrate that ∀x∃yRxy does not imply ∃y∀xRxy.
3 For
example, it will work to choose W so that
RW = {(0, 0), (1, 1)}.
6
R00 T
R01 F
R10 F
R11 . That is to say, T
1
Atomic formulas
As I’ve urged, formulas which contain free variables represent statements which are incomplete. Therefore, we will define truth only for formulas which are closed. The simplest kind of closed formula consists of an n-place predicate followed by a list of n constants. For example, consider R. Intuitively, this says that a bears to b the relation R. What is it for a world to make that happen? What is it for a world to make things stand in some definite way? Well, recall that a world supplies • a domain, determining which objects exist, and • an extension for each predicate, determining how those things are. For example, a world may look like ∗ : a : W= F : R:
this: {0, 1, 2} 2 {0, 1} {h0, 1i, h1, 0i, h2, 2i}
We want to say that the formula R is true in W, because h0, 1i belongs to the W-extension of R. Likewise F a is false in W because in W, a represents 2 and 2 does not belong to the W-extension of F . 1
Before giving the general definition, a couple of niceties. First, if P is a predicate, I will use this notation: • PW is the W-extension of P. Second, if a is a term, then • aW is the object a stands for in W. Recall that I have been using ordinary numerals 0, 1, 2, . . . to represent numbers, and old-fashioned letters , , to represent their names. We adopt the convention that in every possible world, W = 0 W = 1 W = 2 and so on. Now the definition of truth for closed atomic formulas is this: • A formula Pa is true in W just in case aW belongs to PW . • A formula Qab is true in W just in case (aW , bW ) belongs to QW .
2
Truth-functional combinations
So as you can see, the handling of atomic formulas is pretty straightforward. So is the handling of truth-functions. It carries over exactly from the truth-tables of truth-functional logic. So in this case, the truth or falsehood of a formula is perfectly determined by the truth or falsehood of its simpler parts. Let’s restate the procedure in the new language. Suppose that P and Q are formulas of FOL. Then • W |= ¬P iff W 6|= P • W |= P ∧ Q iff both W |= P and W |= Q; • W |= P ∨ Q iff either W |= P, or W |= Q (or both); • W |= P → Q iff either W 6|= P or W |= Q (or both); • W |= P ↔ Q iff either W |= P and W |= Q, or W 6|= P and W 6|= Q. For example, let W be defined as above. Then have we W |= (F a → F b) ↔ ¬Rba? To settle this question, first apply the definition of truth for closed atomic formulas to determine the truth or falsehood at W of F a, F b, and Rba. Then use the truth-tables to chase the truth-values up the formation tree.
3
Quantification
The treatment of quantification is quite a bit more interesting.1 Consider, for example, the formula ∀xF x. 1 The first solution to this problem was given by Alfred Tarski in 1933, “The concept of truth in formalized languages”. We’ll follow an approach from Joseph Shoenfield’s Mathematical Logic (1967).
2
Our earlier strategy was to reduce the question of its truth-value to the question of truth-values of its simpler parts. However, the only simpler subformula of ∀xF x is F x. And in F x, the x occurs free. This means that F x is analogous to a sentence like “it is green” where the “it” is unspecified. Being not closed, the subformula F x ought not to have a truth-value at all. Thus, quantified formulas are not in general built from simpler closed formulas. So, we need a new idea. Suppose again that the domain of W is {0,1,2}. Then what should W be like in order that it make true a universally quantified formula ∀xA? Intuitively, that formula says “everything has the property expressed by A”, or that every object satisfies A. Now according to W, all the things that exist are simply 0, 1, and 2. So, the formula ∀xA should be true over W provided that each of 0, 1 and 2 satisfies A. So, the question we need to answer is this: • what is it for an object to satisfy a formula over a structure? Well, suppose that A is a formula with the one free variable x. Then, • the object o satisfies A over W provided that the result of plugging in for x a name of o is true in W. For example, 0 satisfies F x just in case F is true, and 2 satisfies ∃y∀z(Rxy → Rxz) just in case ∃y∀z(Ry → Rz) is true. Finally the definition of truth for quantified formulas is this: • ∀xA is true in W iff every element of the domain of W satisfies A. • ∃xA is true in W iff some element of the domain of W satisfies A.
4
The method of truth-functional expansion
In truth-functional logic, the definition of truth is easy to apply: just chase the truth-value up the formation tree. Unfortunately, the situation is not so straightforward in first-order logic. In particular, if the domain is infinite, then the truth-value of a generalization depends on the truth-value of infinitely many simpler formulas.2 Luckily, our main purpose is the assessment of deductive arguments. An argument, as ever, is invalid if it has a countermodel—that is, if there’s a world where all premises are true while the conclusion is false. And it turns out that most invalid arguments you’ll ordinarily encounter have countermodels which are very small, with just two or three elements in their domain. Let’s now develop a practical procedure for evaluating formulas in small structures. The key idea is that once the domain of a structure has been fixed as some finite set of objects, universal and existential quantifiers can be regarded as 2 For example, the natural numbers, together with addition, multiplication, and exponentiation can be considered to form a structure. Problems of classical number theory are simply questions whether a given formula is true in that structure.
3
conjunctions and disjunctions. That is, suppose that the elements of the domain are o1 , . . . , ok . We’ll associate, to each quantified formula A, another formula Ao1 ,...,ok called the truth-functional expansion of A over o1 , . . . , ok . Over any structure whose domain contains just o1 , . . . , ok , the formulas A and Ao1 ,...,ok will have the same truth-value. So over a given finite structure, the method of truth-functional expansion reduces the evaluation of a quantified formula to the evaluation of an unquantified one. To state the method, it will help to have one further piece of terminology. Say that an W-instance of a generalization ∀xA or ∃xA is a formula A[x/o], where o is an element of ∗W . For example, if ∗W contains 1, then an instance of ∃x∃y(F y ∧ F x) is ∃y(F y ∧ F 1). Now the method for computing an expansion of a formula over a domain {o1 , . . . , ok } is simply this. • The expansion of a universal generalization is the conjunction of the expansions of its instances • The expansion of an existential generalization is the disjunction of the expansions of its instances • The expansion of a truth-functional combo of some formulas is just that combo of the expansions of those formulas. • The expansion of an atomic formula is just that formula. Examples. For example, suppose that 0, 1, 2 are the elements of ∗W . And consider a random formula A, like ∃xF x → ∀xF x. The expansion A0,1,2 of A over 0, 1, 2 now gets computed like this: A0,1,2
(∃xF x → ∀xF x)0,1,2 (∃xF x)0,1,2 → (∀xF x)0,1,2 (F 0 ∨ F 1 ∨ F 2) → (F 0 ∧ F 1 ∧ F 2).
We can use the expansion of A to compute the truth-value of A. Suppose, for example, that the W-extension of F is {0, 2}. Granted that ∗W is {0, 1, 2}, this reduces the relevant portion of W to the row of a truth-table! Namely, F0 T
F1 F
F2 . T
So it is clear that W |= F 0 ∨ F 1 ∨ F 2. Meanwhile, W 6|= F 0 ∧ F 1 ∧ F 2. Consequently, W 6|= (F 0 ∨ F 1 ∨ F 2) → (F 0 ∧ F 1 ∧ F 2). 4
And that’s just to say W 6|= (∃xF x → ∀xF x)0,1,2 . Since, however, 0, 1, 2 are precisely the elements of W, it follows that A0,1,2 must, over W, have the same truth-value as A! Therefore, W 6|= ∃xF x → ∀xF x. Caution! It should be clear that first-order logic is much more expressive than truth-functional logic. This is because the quantifiers are a fundamentally new concept. Quantifiers let us say things we can’t say only with truth-functions. Consider, for example, the formula ∀xF x. This is certainly not equivalent to a conjunction of its instances. For no matter how many objects you consider, it’s logically possible for there to exist some other object besides those, and this object might not satisfy F x. Thus, ∀xF x does not follow from any conjunction of atomic formulas. For this reason, it should be clear that a quantified formula is not equivalent to its truth-functional expansion. Rather, the method says only that a formula and its expansion over a domain ∗W have the same truth-value over W. To bring this out, here are a couple of exercises. (i) Find a formula, together with two domains, such that the expansion-of-the-formula-over-one domain is not equivalent to its expansion-over-the-other-domain. (ii) Find a formula and a domain such that the formula does not imply its expansion-over-that-domain. Illustrating the method of expansions. The method of truth-functional expansion is useful for distinguishing the meanings of more complicated formulas. Consider, for example, ∀x∃yRxy and ∃y∀xRxy. Are these equivalent? If so, then they must have the same truth-value over all structures whatsoever. Do they? Consider first a structure whose domain is just {0}. In that case, the two formulas both expand into R00. So they must have the same truth-value over all one-element structures. But consider instead the domain {0, 1}. Then the formulas expand like this. On the one hand, (∀x∃y(Rxy))0,1
(∃y(R0y))0,1 ∧ (∃y(R1y))0,1 (R00 ∨ R01) ∧ (R10 ∨ R11).
On the other hand, (∃y∀x(Rxy))0,1
(∀x(Rx0))0,1 ∨ (∀x(Rx1))0,1 (R00 ∧ R10) ∨ (R01 ∧ R11).
Can you find a structure with domain {0, 1} which makes one true and the other false? Finding such a structure amounts to filling in the question marks in R00 ?
R01 ?
R10 ? 5
R11 ?
This is an exercise in truth-functional logic. You should be able to find a W with ∗W = {0, 1} so that W |= ∀x∃yRxy while W 6|= ∃y∀xRxy.3 This will demonstrate that ∀x∃yRxy does not imply ∃y∀xRxy.
3 For
example, it will work to choose W so that
RW = {(0, 0), (1, 1)}.
6
R00 T
R01 F
R10 F
R11 . That is to say, T
